Worksheet: Geometric Sequences

In this worksheet, we will practice calculating the common ratio, find next terms in an geometric sequence, and check if the sequence increases or decreases.

Q1:

Find the common ratio of the geometric sequence 𝑎=1156,152,352,952,2752.

A2

B3

C13

D8

Q2:

Find the next term of the geometric sequence −5,−54,−516,−564,.

A−51,024

B−5,120

C−1,280

D−5256

Q3:

Find the next four terms in the geometric sequence 1165,155,355,….

A155,1165,1495,11,485

B955,2755,8155,24355

C455,111,655,755

D955,8155,2755,24355

Q4:

Find the fifth term of the geometric sequence 186,−143,243,….

A1786

B843

C−443

D−1643

Q5:

State whether the following is true or false: The terms of a geometric sequence can be plotted as a set of collinear points.

Afalse

Btrue

Q6:

Find the next term in the sequence 6,30,150,750,.

Q7:

Which of the following is a geometric sequence?

A𝑎=𝑛(𝑛+2), where 𝑛≥1.

B𝑎=𝑛3, where 𝑛≥2.

C𝑎=3(𝑛+3), where 𝑛≥1.

D𝑎=5𝑎, where 𝑛≥2.

Q8:

State whether the following is true or false:
A geometric sequence is decreasing if its common ratio
𝑟∈(−1,0).

ATrue

BFalse

Q9:

Find the common ratio of a geometric sequence given the middle terms are 56 and 168 respectively.

A112

B37

C73

D13

E3

Q10:

State whether the following is true or false:
A geometric sequence is alternating if its common ratio 𝑟 satisfies 𝑟∈(−1,0).

Afalse

Btrue

Q11:

For an increasing geometric sequence with first term 𝑎, and common ratio 𝑟, which of the following could be true?

A𝑎<−1, −1<𝑟<0

B𝑎>0, 0<𝑟<1

C𝑎<0, −1<𝑟<0

D𝑎<0, 0<𝑟<1

E𝑎>0, −1<𝑟<0

Q12:

A geometric sequence has a first term, 𝑎, and a common ratio, 𝑟. Which of the following conditions ensures that the sequence is NOT alternating?

A𝑎>0, 0<𝑟<1

B𝑎>0, −1<𝑟<0

C𝑎<0, −1<𝑟<0

D𝑎>0, 𝑟=−1

E𝑎>0, 𝑟<−1

Q13:

Find the common ratio of the geometric sequence which satisfies the relation
𝑎=98𝑎, where 𝑛≥1.

A98

B−18

C178

D89

Q14:

The table shows the number of bacteria in a laboratory experiment across four consecutive
days. The number of bacteria can be described by a geometric sequence. Find the common ratio
of this sequence.

Day

1st

2nd

3rd

4th

Number of Bacteria

643

2,572

10,288

41,152

A83

B14

C8

D4

E3

Q15:

Which of the following is not a geometric sequence?

A𝑤7𝑥,−16,7𝑥36𝑤,−49𝑥216𝑤,⋯

B(11,−44,176,−704,…)

C𝑏,𝑏,𝑏,𝑏,…loglogloglog

D119,−157,1171,−1513,⋯

Q16:

For the given sequence, what is the missing term? −60,,−2,160,12,960,−77,760,…

Q17:

Find the value of 𝑚 given the geometric sequence
−4,𝑚,2𝑚+3,….

A−6
or
8

B6
or
−2

C−6
or
2

D−6
or−2

E−2
or
8

Q18:

The table below represents the salary of an employee in three consecutive years in LE. The salary can be described by a geometric sequence.
Find the salary of the employee in the fourth and fifth year, expressed by 𝑎 and 𝑎 respectively.

Year

First

Second

Third

Fourth

Fifth

Salary in LE

673

2,692

10,768

A𝑎=43,072LE, 𝑎=10,768LE

B𝑎=43,072LE, 𝑎=172,288LE

C𝑎=172,288LE, 𝑎=689,152LE

D𝑎=43,072LE, 𝑎=689,152LE

Q19:

Find the value of the second term of the geometric sequence
𝑎=16×2, where 𝑛≥1.

A83

B13,888

C163

D323

Q20:

Find 𝑥 and 𝑦 given the geometric sequence
(1,4𝑥,4𝑦,64,…).

A𝑥=1, 𝑦=64

B𝑥=64, 𝑦=1

C𝑥=164, 𝑦=14,096

D𝑥=1, 𝑦=4

E𝑥=4, 𝑦=1

Q21:

Find the first five terms of the sequence with general term 𝑎=5𝑎, where 𝑛≥1 and 𝑎=2.

A2,10,50,250,1,250

B1,250,250,50,10,2

C10,50,250,1,250,6,250

D10,2,1,250,50,250

Q22:

Find the first five terms of the sequence 𝑎, given 𝑎=14𝑎, 𝑛≥1, and 𝑎=−27.

A−274,−2716,−2764,−27256,−271,024

B−27,−274,−2716,−2764,−27256

C274,2716,2764,27256,271,024

D27,274,2716,2764,27256

Q23:

A geometric sequence is a list of terms which can be written in the form
𝑎,𝑎𝑟,𝑎𝑟,𝑎𝑟,…, where 𝑎 is the first
term and 𝑟 is the common ratio (the number you multiply one term by to get the next term in
the sequence, 𝑟≠1).

Identify 𝑎 and 𝑟 in the following sequence: 250,50,10,2,….

A𝑎=250, 𝑟=15

B𝑎=250, 𝑟=5

C𝑎=50, 𝑟=5

D𝑎=200, 𝑟=45

E𝑎=50, 𝑟=10

Q24:

A geometric sequence is a list of terms which can be written in the form
𝑎,𝑎𝑟,𝑎𝑟,𝑎𝑟,…, where
𝑎 is the first term and 𝑟 is the common ratio (the
number you multiply one term by to get the next term in the sequence, 𝑟≠1).

Identify 𝑎 and 𝑟 in the following sequence: 4,12,36,108,….

A𝑎=3, 𝑟=4

B𝑎=8, 𝑟=4

C𝑎=4, 𝑟=3

D𝑎=2, 𝑟=3

E𝑎=4, 𝑟=8

Q25:

The table below represents the salary of an employee in three consecutive years. The salary
can be described by a geometric sequence. Find the total salary of the employee over 5 years.